Ukraine is currently under martial law and has significant restrictions on speech; it's (justifiably) illegal, for example, to report on troop positions. Starlink isn't changing that; it just provides internet access, filtered through a local ground station subject to any local laws and filtering that are applicable. China has internet access, but not free speech. Same for Russia.

The ground stations aren't in Ukraine. So, Ukraine doesn't have a means of enforcing whatever restrictions on speech come with martial law.